Subjective probability and delay.

Human subjects indicated their preference between a hypothetical $1,000 reward available with various probabilities or delays and a certain reward of variable amount available immediately. The function relating the amount of the certain-immediate reward subjectively equivalent to the delayed $1,000 reward had the same general shape (hyperbolic) as the function found by Mazur (1987) to describe pigeons' delay discounting. The function relating the certain-immediate amount of money subjectively equivalent to the probabilistic $1,000 reward was also hyperbolic, provided that the stated probability was transformed to odds against winning. In a second experiment, when human subjects chose between a delayed $1,000 reward and a probabilistic $1,000 reward, delay was proportional to the same odds-against transformation of the probability to which it was subjectively equivalent.     

Physical and cognitive effort discounting of hypothetical monetary rewards

Effort discounting refers to the decrease in the subjective value of a reward as the effort required to obtain the reward increases. The main aims of this study were to ascertain whether the amount of the reward affects the steepness of the effort discounting process for hypothetical monetary rewards, to identify whether this steepness depends on the type of effort that is required, and to determine whether the steepness of different types of effort covary at the individual level. Two types of effort were studied under hypothetical choice situations: physical effort and cognitive effort. Both physical and cognitive effort discounting were well described by the hyperbolic model. Large rewards were discounted less steeply than small rewards for both types of effort. This finding agrees with the results of prior studies which have found that larger rewards have greater motivational power. In addition, the steepness of physical effort discounting was positively correlated with the steepness of cognitive effort discounting, which suggests that the effort discounting process is a trait‐like characteristic within an individual.     

Modeling Myopic Decisions: Evidence for Hyperbolic Delay-Discounting within Subjects and Amounts

Hyperbolic and exponential discounting functions were compared as models of subjects′ present valuations of delayed rewards. Previous comparisons have been limited by relying on the assumption that discounting rate is independent of reward size; we avoided this limitation by making all comparisons within reward sizes. In Experiment 1, using real rewards in a simulated auction, and in Experiment 2, using hypothetical rewards, we offered subjects five monetary rewards at six delays each and asked them to indicate the smallest amount that they would accept immediately in exchange for those rewards. Both discounting functions were then fit to the six reported amounts for each reward using nonlinear regressions. In both experiments, although both functions fit the data very well, the hyperbolic function fit better for all of the delayed rewards. Furthermore, the hyperbolic function better described the data for 20 of 21 and 14 of 18 subjects in Experiments 1 and 2, respectively.     

Delay-discounting probabilistic rewards: Rates decrease as amounts increase

The independence of delay-discounting rate and monetary reward size was tested by offering subjects (N = 621) a series of choices between immediate rewards and larger, delayed rewards. In contrast to previous studies, in which hypothetical rewards have typically been employed, subjects in the present study were entered into a lottery in which they had a chance of actually receiving one of their choices. The delayed rewards were grouped into small ($30–$35), medium ($55–$65), and large amounts ($70–$85). Using a novel parameter estimation procedure, we estimated discounting rates for all three reward sizes for each subject on the basis of his/her pattern of choices. The data indicated that the discounting rate is a decreasing function of the size of the delayed reward (p < .0001), whether hyperbolic or exponential discounting functions are assumed. In addition, a reliable gender difference was found (p = .005), with males discounting at higher rates than females, on average.     

Delay, Probability, and Social Discounting in A Public Goods Game

A human social discount function measures the value to a person of a reward to another person at a given social distance. Just as delay discounting is a hyperbolic function of delay, and probability discounting is a hyperbolic function of odds-against, social discounting is a hyperbolic function of social distance. Experiment 1 obtained individual social, delay, and probability discount functions for a hypothetical $75 reward; participants also indicated how much of an initial $100 endowment they would contribute to a common investment in a public good. Steepness of discounting correlated, across participants, among all three discount dimensions. However, only social and probability discounting were correlated with the public-good contribution; high public-good contributors were more altruistic and also less risk averse than low contributors. Experiment 2 obtained social discount functions with hypothetical $75 rewards and delay discount functions with hypothetical $1,000 rewards, as well as public-good contributions. The results replicated those of Experiment 1; steepness of the two forms of discounting correlated with each other across participants but only social discounting correlated with the public-good contribution. Most participants in Experiment 2 predicted that the average contribution would be lower than their own contribution.     

Tests of an indifference rule in variable‐delay and double‐reward choice procedures with humans

Four hundred and fifty participants were recruited from Amazon Mechanical Turk across 3 experiments to test the predictions of a hyperbolic discounting equation in accounting for human choices involving variable delays or multiple rewards (Mazur, 1984, 1986). In Experiment 1, participants made hypothetical choices between 2 monetary alternatives, 1 consisting of a fixed delay and another consisting of 2 delays of equal probability (i.e., a variable‐delay procedure). In Experiment 2, participants made hypothetical monetary choices between a single, immediate reward and 2 rewards, 1 immediate and 1 delayed (i.e., a double‐reward procedure). Experiment 3 also used a double‐reward procedure, but with 2 delayed rewards. Participants in all 3 experiments also completed a standard delay‐discounting task. Finally, 3 reward amounts were tested in each type of task ($100, $1000, and $5000). In the double‐reward conditions (Experiments 2 and 3), the results were in good qualitative and quantitative agreement with Mazur's model (1984, 1986). In contrast, when participants made choices involving variable delays (Experiment 1), there was relatively poor qualitative and quantitative agreement with this model. These results, along with our previous findings, suggest the structure of questions in hypothetical tasks with humans can be a strong determinant of the choice pattern.     

Temporal discounting predicts student responsiveness to exchange delays in a classroom token system

Typical assessments of temporal discounting involve presenting choices between hypothetical monetary outcomes. Participants choose between smaller immediate rewards and larger delayed rewards to determine how the passage of time affects the subjective value of reinforcement. Few studies, however, have compared such discounting to actual manipulations of reward delay. The present study examined the predictive validity of a temporal discounting procedure developed for use with children. Forty-six sixth-grade students completed a brief discounting assessment and were then exposed to a classwide intervention that involved both immediate and delayed reinforcement in a multiple baseline design across classrooms. The parameters derived from two hyperbolic models of discounting correlated significantly with actual on-task behavior under conditions of immediate and delayed exchange. Implications of temporal discounting assessments for behavioral assessment and treatment are discussed.     

Integrated versus segregated accounting and the magnitude effect in temporal discounting

Temporal discounting rates in humans generally decrease as the amount of reward increases, a phenomenon known as themagnitude effect. In the present study, we examined whether temporal discounting and the magnitude effect are related to segregation of choices in terms of gains or losses for waiting for or expediting receipt of a reward. Subjects (N = 24) responded to a series of hypothetical choices about amounts of money available either immediately or after a delay. The immediate and delayed amounts either were presented as integrated amounts in the baseline condition or were segregated as differential gains or losses for choosing delayed or expedited consumption (delay and speedup conditions, respectively). Temporal discounting rates decreased in the segregated conditions, in accord with the standard discounted utility model but contrary to the hypothesis that the subjects were choosing on the basis of reward differentials in the baseline condition. The size of the magnitude effect was comparable in the baseline and the delay conditions but decreased in the speed-up condition. These results challenge explanations of the magnitude effect in terms of an increasing proportional sensitivity property of the utility function (Loewenstein & Prelec, 1992) and the hypothesis that subjects choose on the basis of differentials even when the rewards are presented as integrated amounts.     

Effects of reward magnitude frames on measures of delay discounting in a hypothetical money scenario

The current study analyzed the effects of three frames of reward magnitude—quantity, volume, and duration—on the rate at which college students discounted hypothetical, delayed monetary rewards. Hypothetical scenarios were presented using the fill-in-the-blank discounting questionnaire and participants made choices between immediate and delayed hypothetical monetary rewards. Scenarios framed the monetary choices as (a) quantity of dollar bills, (b) height (inches) of a stack of dollar bills, and (c) duration of time spent in a hypothetical cash machine to collect dollar bills. For each scenario, participants' subjective values were used to calculate the area under the curve (AuC). Framing resulted in a moderate effect size: The duration frame yielded significantly smaller AuC values compared to the quantity and volume frames. Thus, the framing of reward magnitude was a significant variable in controlling discounting rates for hypothetical, delayed monetary rewards. Subsequent investigations should be aware of the independent effects of the reward magnitude frames on delay discounting rates.     

DISCOUNTING OF DELAYED FOOD REWARDS IN PIGEONS AND RATS: IS THERE A MAGNITUDE EFFECT?

Temporal discounting refers to the decrease in the present, subjective value of a reward as the time to its receipt increases. Results from humans have shown that a hyperbola-like function describes the form of the discounting function when choices involve hypothetical monetary rewards. In addition, magnitude effects have been reported in which smaller reward amounts are discounted more steeply than larger amounts. The present research examines the cross-species generality of these findings using real rewards, namely food pellets, with both pigeons and rats. As with humans, an adjusting amount procedure was used to estimate the amount of immediate reward judged equal in value to a delayed reward. Different amounts of delayed food rewards (ranging from 5 to 32 pellets in pigeons and from 5 to 20 pellets in rats) were studied at delays varying from 1 s to 32 s. A simple hyperbola, similar to the hyperbola-like mathematical function that describes the discounting of hypothetical monetary rewards in humans, described the discounting of food rewards in both pigeons and rats. These results extend the generality of the mathematical model of discounting. Rates of discounting delayed food rewards were higher for pigeons than for rats. Unlike humans, however, neither pigeons nor rats showed a reliable magnitude effect: Rate of discounting did not vary systematically as a function of the amount of the delayed reward.     

Syntax for calculation of discounting indices from the monetary choice questionnaire and probability discounting questionnaire

The 27‐item Monetary Choice Questionnaire (MCQ; Kirby, Petry, & Bickel, 1999) and 30‐item Probability Discounting Questionnaire (PDQ; Madden, Petry, & Johnson, 2009) are widely used, validated measures of preferences for immediate versus delayed rewards and guaranteed versus risky rewards, respectively. The MCQ measures delayed discounting by asking individuals to choose between rewards available immediately and larger rewards available after a delay. The PDQ measures probability discounting by asking individuals to choose between guaranteed rewards and a chance at winning larger rewards. Numerous studies have implicated these measures in addiction and other health behaviors. Unlike typical self‐report measures, the MCQ and PDQ generate inferred hyperbolic temporal and probability discounting functions by comparing choice preferences to arrays of functions to which the individual items are preconfigured. This article provides R and SPSS syntax for processing the MCQ and PDQ. Specifically, for the MCQ, the syntax generates k values, consistency of the inferred k, and immediate choice ratios; for the PDQ, the syntax generates h indices, consistency of the inferred h, and risky choice ratios. The syntax is intended to increase the accessibility of these measures, expedite the data processing, and reduce risk for error.     

Social discounting and delay discounting

Social discounting was measured as the amount of money a participant was willing to forgo to give a fixed amount (usually $75) to another person. In the first experiment, amount forgone was a hyperbolic function of the social distance between the giver and receiver. In the second experiment, degree of social discounting was an increasing function of reward magnitude whereas degree of delay discounting was a decreasing function of reward magnitude. In the third experiment, the shape of the function relating delayed rewards to equally valued immediate rewards for another person was predicted from individual delay and social discount functions. All in all, the studies show that the social discount function, like delay and probability discount functions, is hyperbolic in form. Copyright © 2007 John Wiley & Sons, Ltd.     

Within-subject comparison of real and hypothetical money rewards in delay discounting

A within-subject design, using human participants, compared delay discounting functions for real and hypothetical money rewards. Both real and hypothetical rewards were studied across a range that included $10 to $250. For 5 of the 6 participants, no systematic difference in discount rate was observed in response to real and hypothetical choices, suggesting that hypothetical rewards may often serve as a valid proxy for real rewards in delay discounting research. By measuring discounting at an unprecedented range of real rewards, this study has also systematically replicated the robust finding in human delay discounting research that discount rates decrease with increasing magnitude of reward. A hyperbolic decay model described the data better than an exponential model.     

Delay or probability discounting in a model of impulsive behavior: effect of alcohol

Little is known about the acute effects of drugs of abuse on impulsivity and self-control. In this study, impulsivity was assessed in humans using a computer task that measured delay and probability discounting. Discounting describes how much the value of a reward (or punisher) is decreased when its occurrence is either delayed or uncertain. Twenty-four healthy adult volunteers ingested a moderate dose of ethanol (0.5 or 0.8 g/kg ethanol: n = 12 at each dose) or placebo before completing the discounting task. In the task the participants were given a series of choices between a small, immediate, certain amount of money and $10 that was either delayed (0, 2, 30, 180, or 365 days) or probabilistic (i.e., certainty of receipt was 1.0, .9, .75, .5, or .25). The point at which each individual was indifferent between the smaller immediate or certain reward and the $10 delayed or probabilistic reward was identified using an adjusting-amount procedure. The results indicated that (a) delay and probability discounting were well described by a hyperbolic function; (b) delay and probability discounting were positively correlated within subjects; (c) delay and probability discounting were moderately correlated with personality measures of impulsivity; and (d) alcohol had no effect on discounting.     

PREFERENCE REVERSALS DUE TO MYOPIC DISCOUNTING OF DELAYED REWARD

Abstract— A basic stationarity axiom of economic theory assumes stable preference between two deferred goods separated by a fixed time. To test this assumption, we offered subjects choices between delayed rewards, while manipulating the delays to those rewards. Preferences typically reversed with changes in delay, as predicted by hyperbolic discounting models of impulsiveness. Of 36 subjects, 34 reversed preference from a larger, later reward to a smaller, earlier reward as the delays to both rewards decreased. We conclude that the stationarity axiom is not appropriate in models of human choice.     

A Measure of Delay Discounting Within the Academic Domain

It is important to better understand the decision‐making processes involved in student procrastination, in order to develop interventions that reduce this common problem. Students may procrastinate because studying produces delayed reinforcers; however, no task measuring delay discounting of academic outcomes currently exists. In Experiment 1, we developed and piloted a measure of academic discounting modeled on titrating‐amount tasks successfully used in the discounting literature. Participants made hypothetical choices between working for money (the smaller, sooner reinforcer) and working on an assignment that was due at various times (the larger, later reinforcer). Participants showed systematic decreases in the subjective value of the assignment as a function of delay, and the hyperbolic and hyperboloid models described the shape of this decrease in value well. In general, larger delayed rewards are discounted less steeply than smaller delayed rewards (the magnitude effect). In Experiment 2, we observed the magnitude effect in academic discounting: Participants discounted a “not important” assignment more steeply than an “important” assignment. In the hyperboloid model, this change was captured by an increase in the s parameter. Results provide support for the validity of the academic discounting task. Copyright © 2018 John Wiley & Sons, Ltd.     

Comparing rapid assessments of delay discounting with real and hypothetical rewards in children

Recent advances in assessment methodology have resulted in a highly efficient procedure for obtaining delay discounting rates for adults: a 5‐trial adjusting delay task (ADT‐5) examining intertemporal choice for hypothetical rewards. The low participant burden of this task makes it potentially useful for children, with whom delay discounting research is relatively limited. However, it is unknown whether results from this task match choice for real rewards. The present study assessed delay discounting for real and hypothetical monetary rewards using a modified ADT‐5 with 9 children admitted to a psychiatric day treatment program. Participants completed up to 3 tasks with each reward type in alternating order. No difference in discounting rate, via log(k), was observed between the first task of each reward type. This finding was replicated across subsequent tasks for the subset of participants (n = 6) who completed all 6 tasks. However, delay discounting of real and hypothetical rewards was not found to be statistically equivalent. These results suggest that a modified ADT‐5 using hypothetical rewards may be a viable option for assessing delay discounting in children with psychiatric diagnoses, but additional research is needed to explicitly examine whether hypothetical and real rewards are discounted equivalently in this population.     

DISCOUNTING OF DELAYED REWARDS:

Abstract— In this study, children, young adults, and older adults chose between immediate and delayed hypothetical monetary rewards The amount of the delayed reward was held constant while its delay was varied All three age groups showed delay discounting, that is, the amount of an immediate reward judged to be of equal value to the delayed reward decreased as a function of delay The rate of discounting was highest for children and lowest for older adults, predicting a life-span developmental trend toward increased selfcontrol Discounting of delayed rewards by all three age groups was well described by a single function with agesensitive parameters (all R²s>94) Thus, even though there are quantitative age differences in delay discounting, the existence of an age-invanant form of discount function suggests that the process of choosing between rewards of different amounts and delays is qualitatively similar across the life span